Problem: Simplify the following expression and state the condition under which the simplification is valid. $z = \dfrac{t^3 + 3t^2 - 54t}{t^3 + 12t^2 + 27t}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ z = \dfrac {t(t^2 + 3t - 54)} {t(t^2 + 12t + 27)} $ $ z = \dfrac{t}{t} \cdot \dfrac{t^2 + 3t - 54}{t^2 + 12t + 27} $ Simplify: $ z = \dfrac{t^2 + 3t - 54}{t^2 + 12t + 27}$ Since we are dividing by $t$ , we must remember that $t \neq 0$ Next factor the numerator and denominator. $ z = \dfrac{(t + 9)(t - 6)}{(t + 9)(t + 3)}$ Assuming $t \neq -9$ , we can cancel the $t + 9$ $ z = \dfrac{t - 6}{t + 3}$ Therefore: $ z = \dfrac{ t - 6 }{ t + 3 }$, $t \neq -9$, $t \neq 0$